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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remar ..."
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Cited by 169 (52 self)
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This paper reviews the Fourierseries method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourierseries method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...
Performance Modeling of Distributed Memory Architectures
 JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING
, 1991
"... We provide performance models for several primitive operations on data structures distributed over memory units interconnected by a Boolean cube network. In particular, we model single source, and multiple source concurrent broadcasting or reduction, concurrent gather and scatter operations, shifts ..."
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Cited by 20 (7 self)
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We provide performance models for several primitive operations on data structures distributed over memory units interconnected by a Boolean cube network. In particular, we model single source, and multiple source concurrent broadcasting or reduction, concurrent gather and scatter operations, shifts along several axes of multidimensional arrays, and emulation of butterfly networks. We also show how the processor configuration, data aggregation, and the encoding of the address space affect the performance for two important basic computations: the multiplication of arbitrarily shaped matrices, and the Fast Fourier Transform. We also give an example of the performance behavior for local matrix operations for a processor with a single path to local memory, and a set of registers. The analytic models are verified by measurements on the Connection Machine model CM2.
Accuracy Enhancement for Higher Derivatives using Chebyshev Collocation and a Mapping Technique
, 1994
"... We study a new method in reducing the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and TalEzer, and the proper choice of the parameter ff, the roundoff error of the kth derivative can be reduced from O(N 2k ) to O((N jln ..."
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Cited by 19 (1 self)
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We study a new method in reducing the roundoff error in computing derivatives using Chebyshev collocation methods. By using a grid mapping derived by Kosloff and TalEzer, and the proper choice of the parameter ff, the roundoff error of the kth derivative can be reduced from O(N 2k ) to O((N jln fflj) k ), where ffl is the machine precision and N is the number of collocation points. This drastic reduction of roundoff error makes mapped Chebyshev methods competitive with any other algorithm in computing second or higher derivatives with large N . We also study several other aspects of the mapped Chebyshev differentiation matrix. We find that 1) the mapped Chebyshev methods requires much less than ß points to resolve a wave, 2) the eigenvalues are less sensitive to perturbation by roundoff error, and 3) larger time steps can be used for solving PDEs. All these advantages of the mapped Chebyshev methods can be achieved while maintaining spectral accuracy. 1 Introduction In [5], we...
All Right. Reserved by Academic Press, New York and London Printed in Belgium A Numerical Study of 20 Turbulence
, 1976
"... A simulation of 2D turbulence in a square region with periodic boundary conditions has been performed using a highly accurate approximation of the inviscid NavierStokes equations to which a modified viscosity has reen added. A series of flow pictures show how a random initial vorticity distributio ..."
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A simulation of 2D turbulence in a square region with periodic boundary conditions has been performed using a highly accurate approximation of the inviscid NavierStokes equations to which a modified viscosity has reen added. A series of flow pictures show how a random initial vorticity distribution quickly a.ssumes a stringlike pattern which persists as the flow simplifies into a few &quot;cyclones &quot; or &quot;finite area vortex regions&quot;. This trend towards welldefined largescale structures can make it questionable if the 2D flow should be described as &quot;turbulent &quot; and it casts some doubts on the concept of inertial range and the relevance of energy spectra. The change in appearance seems to be associated with a buildup of phase correlations in the Fourier representation of the vorticity field. During this initial buildup, the energy spectrum seems to follow a ksIaw, but this behavior does not persist. If there is a power law for steady turbulence the results suggest that is more likely to be a k'law.
Parallel Implementation of a DataTranspose Technique for the Solution of Poisson's Equation in Cylindrical Coordinates
"... We present a parallel finitedifference algorithm for the solution of the 3D cylindrical Poisson equation. The algorithm is based on a datatranspose technique, in which all computations are performed independently on each node, and all communications are restricted to global 3D datatransposition b ..."
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We present a parallel finitedifference algorithm for the solution of the 3D cylindrical Poisson equation. The algorithm is based on a datatranspose technique, in which all computations are performed independently on each node, and all communications are restricted to global 3D datatransposition between nodes. The datatranspose technique aids us in implementing two sequential algorithms for the solution of the cylindrical Poisson equation. The first algorithm is based on the alternating direction implicit (ADI) method and the second is based on Fourier Analysis (FA). In both algorithms we first perform a Fourier transform in the naturally periodic azimuthal coordinate direction. This decouples the 3D problem into a set of independent 2D problems which are then solved by the ADI and FA methods. In the ADI method we convert each 2D problem into two sets of 1D (tridiagonal) problems which are then solved iteratively by alternating in the radial and vertical coordinate directions. In th...
unknown title
"... A finite analytic method for sol advection–diffusion equation wit,*,1, 5800, iversi rm 2 cell (the number of particles) and dividing by the cell volume. Random walk models are representative of this class [2–4]. Lagrangian methods are useful for locating the spatial extent of a contaminant plume or ..."
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A finite analytic method for sol advection–diffusion equation wit,*,1, 5800, iversi rm 2 cell (the number of particles) and dividing by the cell volume. Random walk models are representative of this class [2–4]. Lagrangian methods are useful for locating the spatial extent of a contaminant plume or for deriving concentration fields for plumes of small extent. However under contract #LVLX0006.
Parallel Fast Fourier Transforms for Non Power of Two Data
"... This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation r ..."
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This report deals with parallel algorithms for computing discrete Fourier transforms of real sequences of length N not equal to a power of two. The method described is an extension of existing power of two transforms to sequences with N a product of small primes. In particular, this implementation requires N = 2 p 3 q 5 r . The communication required is the same as for a transform of length N = 2 p . The algorithm presented is intended for use in the solution of partial differential equations, or in any situation in which a large number of forward and backward transforms must be performed and in which the Fourier Coefficients need not be ordered. This implementation is a one dimensional FFT but the techniques are applicable to multidimensional transforms as well. The algorithm has been implemented on a 128 node Intel Ipsc/860. 1 Introduction The algorithm described in this report is an extension of the parallel algorithm given by Swarztrauber [2] and Walker [3] for computing ...